### Abstract

We consider the invariant measure of homogeneous random walks in the quarter-plane. In particular, we consider measures that can be expressed as a countably infinite sum of geometric terms which individually satisfy the interior balance equations. We demonstrate that the compensation approach is the only method that may lead to such a type of invariant measure. In particular, we show that if a countably infinite sum of geometric terms is an invariant measure, then the geometric terms in an invariant measure must be the union of at most six pairwise-coupled sets of countably infinite cardinality each. We further show that for such invariant measure to be a countably infinite sum of geometric terms, the random walk cannot have transitions to the north, northeast or east. Finally, we show that for a countably infinite weighted sum of geometric terms to be an invariant measure at least one of the weights must be negative.

Original language | English |
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Number of pages | 20 |

Journal | Queueing systems |

DOIs | |

Publication status | E-pub ahead of print/First online - 8 Jul 2019 |

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### Keywords

- UT-Hybrid-D
- Compensation approach
- Geometric term
- Invariant measure
- Pairwise-coupled
- Quarter-plane
- Random walk
- Algebraic curve

### Cite this

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**Necessary conditions for the compensation approach for a random walk in the quarter-plane.** / Chen, Yanting; Boucherie, Richard J.; Goseling, Jasper.

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - Necessary conditions for the compensation approach for a random walk in the quarter-plane

AU - Chen, Yanting

AU - Boucherie, Richard J.

AU - Goseling, Jasper

N1 - Springer deal

PY - 2019/7/8

Y1 - 2019/7/8

N2 - We consider the invariant measure of homogeneous random walks in the quarter-plane. In particular, we consider measures that can be expressed as a countably infinite sum of geometric terms which individually satisfy the interior balance equations. We demonstrate that the compensation approach is the only method that may lead to such a type of invariant measure. In particular, we show that if a countably infinite sum of geometric terms is an invariant measure, then the geometric terms in an invariant measure must be the union of at most six pairwise-coupled sets of countably infinite cardinality each. We further show that for such invariant measure to be a countably infinite sum of geometric terms, the random walk cannot have transitions to the north, northeast or east. Finally, we show that for a countably infinite weighted sum of geometric terms to be an invariant measure at least one of the weights must be negative.

AB - We consider the invariant measure of homogeneous random walks in the quarter-plane. In particular, we consider measures that can be expressed as a countably infinite sum of geometric terms which individually satisfy the interior balance equations. We demonstrate that the compensation approach is the only method that may lead to such a type of invariant measure. In particular, we show that if a countably infinite sum of geometric terms is an invariant measure, then the geometric terms in an invariant measure must be the union of at most six pairwise-coupled sets of countably infinite cardinality each. We further show that for such invariant measure to be a countably infinite sum of geometric terms, the random walk cannot have transitions to the north, northeast or east. Finally, we show that for a countably infinite weighted sum of geometric terms to be an invariant measure at least one of the weights must be negative.

KW - UT-Hybrid-D

KW - Compensation approach

KW - Geometric term

KW - Invariant measure

KW - Pairwise-coupled

KW - Quarter-plane

KW - Random walk

KW - Algebraic curve

UR - http://www.scopus.com/inward/record.url?scp=85068816691&partnerID=8YFLogxK

U2 - 10.1007/s11134-019-09622-1

DO - 10.1007/s11134-019-09622-1

M3 - Article

AN - SCOPUS:85068816691

JO - Queueing systems

JF - Queueing systems

SN - 0257-0130

ER -